Complementary local-global approach for phonon mode connectivities

Sorting and assigning phonon branches (e.g., longitudinal acoustic) of phonon modes is important for characterizing the phonon bands of a crystal and the determination of phonon properties such as the Gr\"uneisan parameter and group velocity. To do this, the phonon band indices (including the longitudinal and transverse acoustic) have to be assigned correctly to all phonon modes across a path or paths in the Brillouin zone. As our solution to this challenging problem, we propose a computationally efficient and robust two-stage hybrid method that combines two approaches with their own merits. The first is the perturbative approach in which we connect the modes using degenerate perturbation theory. In the second approach, we use numerical fitting based on least-squares fits to circumvent local connectivity errors at or near exact degenerate modes. The method can be easily generalized to other condensed matter problems involving Hermitian matrix operators such as electronic bands in tight-binding Hamiltonians or in a standard density-functional calculation, and photonic bands in photonic crystals.